Related papers: Test elements in torsion-free hyperbolic groups

Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every…

In this paper, we mainly study hyperbolic semigroups from which we get non-empty escaping set and Eremenko's conjecture remains valid. We prove that if each generator of bounded type transcendental semigroup S is hyperbolic, then the…

We develop methods to control the first-order theory of groups arising as certain direct limits of torsion-free hyperbolic groups, answering several questions in the literature. We construct simple torsion-free Tarski monsters $\Gamma$…

We give an easily checkable algebraic condition which implies that two elements of a finitely generated free group are members of distinct doubly-twisted conjugacy classes with respect to a pair of homomorphisms. We further show that this…

We prove the decidability of the elementary theory of a free group.

If $G_1$ and $G_2$ are torsion-free hyperbolic groups and $P<G_1\times G_2$ is a finitely generated subdirect product, then the conjugacy problem in $P$ is solvable if and only if there is a uniform algorithm to decide membership of the…

We use a recent result of Alexander and Nishinaka to show that if $G$ is a non-elementary torsion-free hyperbolic group and $R$ is a countable domain, then the group ring $RG$ is primitive. This implies that the group ring $KG$ of any…

This is the second in a series of papers about torsion-free groups which act properly and cocompactly on a CAT(0) metric space with isolated flats and relatively thin triangles. Our approach is to adapt the methods of Sela and others for…

We describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group.

Let G be a discrete, torsion free group with a finite dimensional classifying space BG. We show that the existence of a gamma-element for such G is a metric, that is, coarse, invariant of G. We also obtain results for groups with torsion.…

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly…

Let $G$ be an acylindrically hyperbolic group. We prove that if $G$ has no non-trivial finite normal subgroups, then the set of invertible elements is dense in the reduced $C^\ast$-algebra of $G$. The same result is obtained for finite…

We study density of parabolic elements in a finitely generated relatively hyperbolic group $G$ with respect to a word metric. We prove this density to be zero (apart from degenerate cases) and the limit defining the density to converge…

Let $M$ be a cancellative and commutative (additive) monoid. The monoid $M$ is atomic if every non-invertible element can be written as a sum of irreducible elements, which are also called atoms. Also, $M$ satisfies the ascending chain…

In this note we show that any homomorphism from a pure surface braid group to a torsion-free hyperbolic group either has a cyclic image or factors through a forgetful map. This extends and gives a new proof of an earlier result of the…

Let X be a locally compact geodesically complete CAT(0) space and G be a discrete group acting properly and cocompactly on X. We show that G contains an element acting as a hyperbolic isometry on each indecomposable de Rham factor of X. It…

We prove that an automorphism $\phi:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.

We investigate which free constructions (amalgamated products and HNN-extensions) over word hyperbolic groups produce groups that are again word hyperbolic. A complete answer is obtained for the case when the amalgamated subgroups are…

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in…

We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and…